Chapter 03.02
Rekhaank or Rekhaankit ank or Negative digit or Vinculum ank
* Rekhaank or Rekhaankit ank or Negative digit or Vinculum ank
* Converting normal numbers into vinculum and Converse
* Normalizing Vinculum numbers
(1) Ordinary and Vedic numbers:
Generally, all the digits of the numbers we use in mathematics are positive. For example: All the digits 1, 2, 3, 5, 4 and 6 of the number 123546 are positive.
But in Vedic mathematics such numbers are also used whose all digits are either positive or negative or can be of both types i.e. both positive and negative.
Now the question arises that why and how are negative numbers used in Vedic mathematics?
The answer to this question is that all the digits of the numbers used in Vedic mathematics are kept 5 or smaller than 5. Big numbers 6, 7, 8, 9 are not used in Vedic mathematics. How does Vedic mathematics compensate for the lack of these numbers?
Digits greater than five, 6, 7, 8, 9 are converted into smaller digits. In the process of converting numbers larger than five, 6, 7, 8, 9 into smaller numbers, negative numbers are obtained. These negative numbers are called Vinkulam numbers.
Therefore, Vedic numbers include both positive and negative numbers.
1. Normal Digits
If digits are commonly used by us in the form of mathematical digits. So these digits are called normal digits. It starts from zero (0) and goes to infinity (∞).
2. Vedic Digits
Only 14 types of numbers are used in Vedic mathematics. Which are as follows –> 0, 1, 2, 3, 4, 5, 6 or –4, 7 or –3, 8 or –2, 9 or –1 etc.
Hence, in Vedic mathematics these fourteen numbers are 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, –1, –2, –3, –4. We use these in Vedic mathematics.
Vinkulam
In Vedic mathematics, the writing of digits in negative form is called Vinkulam and these digits are called Vinkulam Digits.
In Vedic mathematics, a negative digit (–m) is expressed by putting a negative sign (bar / Line) or dash above the digit m or by a (bar / Line) or dash symbol.
Therefore, in Vedic mathematics, the negative number (–4) is expressed as 4' by putting a negative sign or dash above the number 4, which is called Vinakulam 4' or Binakulam of (–4).
Principle of Vinkulam:
The process of converting numbers 6, 7, 8, 9 larger than five into smaller numbers is called Vinkulam. Due to which they get small and negative marks. These small and negative numbers are called Vinkulam numbers.
As we have already told that in the numbers used in Vedic mathematics, all the digits are kept 5 or smaller than 5. Therefore, all the digits in the number which are greater than 5 (6, 7, 8, 9) are replaced by their Vinkulam digits and appropriate actions are performed.
Other names of Vinkulam digits
We also know Vinukulam digits by the name of line (rekhaank ) or underlined digit (rekhaankit ank ) or negative digit or negative digit (rinaank ) or Vinkulam or dash digit or bar digit etc.
Rekhaank or Rekhaankit ank or Negative digit or Vinculum ank
Rekhank :–Rekhank is a negative number or a digit with a bar on its top, this bae is used to show a negative number.
Rekhaank or Rekhaankit ank or Vinkulam – Theory and Application
It is very important to know the principles of Vinkulam in order to learn Vedic Mathematics. Vinkulam is also known as Rekhaank or Rekhaankit ank or Negative Number or dash digit in Vedic mathematics.
At present, only ten positive numbers (Zero to nine 0, 1, 2, 3, 4, 5, 6, 7, 8, 9 ) are used in addition, subtraction, multiplication and division in mathematics. That is, all numbers are positive in mathematics.
Thus we can say that in Vedic maths we use only 14 following numbers like 0, 1, 2, 3, 4, 5, 6 or –4 or 4' , 7 or –3 or 3' , 8 or –2 or 2', 9 or – 1 or 1' etc.
For Example
* All the digits of the single digit number 1, 3, 7, 6, 5 are positive.
* All the digits of the two-digit number 75 are positive.
* All the digits of the three-digit number 785 are positive.
Similarly, all the digits of the big number number 8273505 are positive.
Although all the numbers from 0 to 9 are used in Vedic mathematics, but for our convenience and to make some calculations accessible, we take them as numbers or numbers less than 5 or 5 only. That is, the big numbers 6, 7, 8, 9 are not used in these calculations.
In the process of converting the numbers 6, 7, 8, 9 into smaller numbers, we get negative numbers. Also, both positive and negative numbers are used in Vedic mathematics.
All digits of numbers used in Vedic maths are kept as 5 or less than 5.
In Vedic Mathematics, writing the digits of numbers in negative form is called Vinkulam. In Vedic mathematics, a negative number (–2) is expressed by putting a negative sign above the number 2 or by a bar symbol. Here we will write it as –2 or 2'. Which will be read Vinkulam two or minus 2.
In the numbers used in Vedic mathematics, all numbers are kept 5 or less than 5. Therefore, whatever digits are greater than 5 (6, 7, 8, 9) in the number, all those (6, 7, 8, 9) digits are replaced by their Vinkulam digits.
Thus we can say that we use vinculum numbers along with natural numbers including zero to make calculations small and simple in Vedic Maths. Thus only 14 types of digits are used in Vedic mathematics. Which are as follows → 0, 1, 2, 3, 4, 5, 6 or –4, 7 or –3, 8 or –2, 9 or –1 etc.
To find Vinkulam number or Vinkulam of a number:
Vinkulam of any number greater than 5 has to be found. Vinkulam is the 'negative deviation from 10' of the larger number.
Since numbers greater than 5, 6, 7, 8 and 9 are all less than 10, their deviations from 10 will have negative signs. These numbers obtained from deviations are displayed by putting a negative sign (–) in front of them or a dash sign (') above them. These underlined numbers are called Vinkulam or Vinkulam numbers.
For Example
To find Vinkulam of number 6.
Deviation of number 6 from 10 = 6 –10 = –4 or 4'
Vinkulam (digit) of number 6 = –4 or 4'
Similarly, Vinkulam (digit) of number 7 = 7–10=–3 or 3'
And
Vinkulam (digit) of number 8 = 8–10=–2 or 2'
Vilakulam (digit) of number 9 =9–10=–1 or 1'
(6) Nikhilam Sutra:
Nikhilam Sutra is –
'Nikhilam Navatah Charam Dashatah' means 'Subtract each digit from 9 and the last right digit from 10.'
With the help of Nikhilam Sutra, Vinakulam of a group of numbers can be easily obtained. Hence, this Nikhilam formula is a powerful means of writing the given ordinary number in the form of Vinkulam Number.
(7) Writing an ordinary number in the form of Vinkulam number:
To write a number in Vinkulum form, the digits greater than 5 in that number are replaced with their Vinkulum digits. And the digits in the number which are less than 5 are written as they are.
Now we will describe in detail the complete method of writing Vinkulam numbers.
To find the vinculum of a number:
If we want to find the vinkulam of a number greater than 5, then its 'deviation from 10' is found. In this way, the negative soul number that comes, is a Vinkulam number.
Important Notes -
If there is no digit on the left side of the number used to find Vinkulam, then zero is written there. i.e. 0+1=1
for example
Write the number 94 or 094 in Vinkulam form.
Vinkulam (digit) of number 9 = •0(9–10)4
By adding 1 to 0 on the left side of the number 9,
0+1=1
Therefore
94=11'4
in short
94 = 094 = (0+1)(9–10)4=1(–1)4=11'4
* To find Vinkulam of number 6.
Deviation of number 6 from 10 = 6 – 10 = –4
Vinkulam of number 6 = overline{4}
* To find Vinkulam of number 7.
Deviation of number 7 from 10 = 7 – 10 = –3
Vinkulam of digit 7 = overline{ 3}
* To find Vinkulam of number 8.
Deviation of number 8 from 10 = 8 – 10 = –2
Vinkulam of digit 8 = overline{2}
Similarly,
* To find Vinkulam of number 9.
Deviation of number 9 from 10 = 9 – 10 = –1
Vinkulam of digit 9 = overline{1}
Thus we find that 0, 1, 2, 3, 4 and 5 do not have vinculum, only 6, 7, 8 and 9 numbers greater than 5 have vinculum.
Vinkulam of number 6 = –4
Vinkulam of number 7 =–3
Vinkulam of number 8 =–2
Vinkulam of number 9 =–1
We used and kept the digits smaller than 5 or in the form of 1, 2, 3, 4, 5 or less than 5 in Vedic maths.
Therefore, whatever digits are greater than 5 (6, 7, 8, 9) in the number, in place of all those (6, 7, 8, 9) digits, their Vinkulam digits are written.
Vinkulam of 6 is –4,
Vinkulam of 7 is –3,
Vinkulam of 8 is –2,
Vinkulam of 9 is –1.
Exercise 1
Find the vinkulams of the following numbers:
1, 9, 2, 8, 3, 7, 4, 6, 5
Ans.
1, –1, 2, –2, 3, –3, 4, –4, 5
Converting ordinary number to Vinkulam number:
To write a number in Vinkulam form, the digits greater than 5 appearing in that number are changed to their Vinkulam digits. Numbers less than 5 are written as they are. Now we will learn the whole method through different examples.
We have learned to make vinukulam digits of common digits of a digit.
Rules for converting two digit normal numbers into Vinculum numbers.
Rule Number 1
Look at the numbers and put the digits greater than 5 in brackets ( ).
Rule Number 2
If there is only one digit in brackets ( ), subtract it from 10 and add 1 to the digit next to it.
Where the Purvotmanka is less than 5 than the other's.
* Write the vinculum of 18.
Sol. 18 = 1(8) = 1(8–10) = •1 (–2) = 2 2'
* Write the vinculum of 27.
Sol. 27 = 2(7) = 2(7–10) = •2(–3) = 3 3'
* Write the vinculum of 46.
Sol. 46 = 4(6) = 4(6–10) = •4(–4) = 5 4'
Type 2
Where the Purvotmanka is more than 5 than the other's.
* Write the vinculum of 89.
Sol.
089 = 0(89)
By Nikhilam Navatah Charam Dashatah Sutra.
= •0(8–9) (9–10)= 1(–1)(–1) = 1 1' 1'
* Write the vinculum of 76.
Sol.
076 = 0(76)
By Nikhilam Navatah Charam Dashatah Sutra.
= •0(7–9) (6–10)= 1(–2)(–4) = 1 2' 4'
Rules for converting three digit ordinary numbers into vinculum numbers.
Rule No. 1
Look at the numbers and write the digits greater than 5 in brackets ( ).
Rule No. 2
If there is only one digit in brackets ( ) then subtract it from 10 and add 1 to the digit next to it.
Type 1 – When only unit digit is greater than five.
For example
Write the vinculum of 318.
Sol.
318 = 31(8) = 31(8–10) = 3•1(–2) = 32 2'
Write the vinculum of 247.
Sol.
247 = 24(7) = 24(7–10) = 2•4(–3) =25 3'
Write the vinculum of 145.
Sol.
146 = 14(6) = 14(6–10) = 1•4(–4) = 15 4'
Type 2 – When only unit digit is greater than five.
Write the vinculum of 189.
Sol.
189 = 1(89)
By Nikhilam Navatah Charam Dashatah Sutra.
= •1(8–9) (9–10)= 2(–1)(–1) = 2 1' 1'
Write the vinculum of 276.
Sol.
276 = 2(76)
By Nikhilam Navatah Charam Dashatah Sutra.
= •2(7–9) (6–10)= 3 2' 4'
Writing the vinculum of big number.
Write the vinculum of 26786.
Sol.
26786 = 2(6786)
खिलम नवत: चरमं दशत: नियम से।
By Nikhilam Navatah Charam Dashatah Sutra.
= •2(6–9) (7–10)(8–9) (6–10)
= 3(–3)(–3)(–2) (–4)
= 3 3' 3' 2' 4'
Write the vinculum of 681786.
681786
= (68)1(786)
= 0(68)1(786)
= •0(68)•1(786)
= 1(6–9)(8–10) 2(7–9)(8–9)(6–10)
= 1(–3)(–2) 2 (–2)(–1)(–4)
= 1 3' 2' 2 2' 1' 4'
1918176 = 21'22'22'4'
Example 2.
Write the number-1918176 in Vinkulam form.
Given number = 1918176=1(9)1(8)1(76)
Here three groups of numbers greater than 5 are formed (9), (8) and (76).
To find the vinculum of group (76):
By taking the deviation of the right side digit 6 by 10 and the deviation of the second digit 7 by 9.
Binakulam of number 6 = –4 or 4'
Vinkulam of number 7 –2 or 2'
To find Vinkulam of group (8):
Taking deviation of number 8 from 10,
Vitakulam of number 8 = –2 or 2'
To find Vinkulam of group (9):
Taking the deviation of the number 9 from 10,
Vinkulam of number 9 = –1 or 1'
Now in place of the digits appearing in each group, their Vitakulam digits will be placed and the value of the digit located on the left side of each group will be increased by one.
Hence in Vinakulam form,
1918176 = 21'22'22'4'
In short the above calculation can be done as follows-
1918176 = 1(9)1(8)1(76)
=(•1) (9-10)(•1)(8-10)(•1) [(7-9)(6-10)]
=2(–1)2(–2)2[(–2)(–4)]
=21' 2 2' 2 2' 4'
Vinkulam numbers : Those numbers in which both positive and negative digits are used are called Vinkulam numbers.
Normal numbers can be easily converted into Vinkulam numbers by Nikhilam formula.
Nikhilam Sutra is Nikhilam Navatah Charam Dashatah.
In other words ‘Subtract each digit from 9 and the last right digit (unit) from 10.’ With the help of this formula, we can easily convert normal numbers into Vinkulam numbers.
B. Conversion of vinculum number into general numbers
The process of conversion is exactly reverse to the already what we have done earlier. Rekhanks are converted by Nikhilam where as other digits by ‘Ekanyunena’ sutra.
ॠणांक (विनकुलम्) संख्या को सामान्य संख्या में बदलना
1. ॠणांक अंक का धनात्मक मान लेते हैं। ॠणांक के ऊपर शिरो रेखा की चिन्ता नही करते हुए चरमं अंक को 10 में से घटाते हैं।
2. निखिलम् अंक को 9 में से घटाते हैं, अथवा इस मान का परम मित्र लिखते हैं।
3. ॠणांक अंक के पूर्व अंक पर एकन्यून चिन्ह लगाते हैं।
* 2 2' का साधारण रूप लिखिए।
* Write the normal number from vinculum 22' .
Sol.
2 2' = *2 (10– 2) = 1(8) = 18
* 3 3' का साधारण रूप लिखिए।
* Write the normal number from vinculum 33'.
Sol.
3 3’ = *3(10 – 3) = 2(7) = 27
* 5 4' का विनकुलम लिखिए।
* Write the normal number from vinculum 54'.
Sol.
5 4' = *5(10 –4 ) = 4(6) = 46
* 4 3' 2' का विनकुलम लिखिए।
हल–
5 4' = *5(10 –4 ) = 4(6) = 46
* Write the normal number from vinculum of 46.
Sol.
5 4' = *5(10 –4 ) = 4(6) = 46
Example:
Example:
Example:
f
Example:
ॠणांक (विनकुलम्) संख्या को सामान्य संख्या में बदलना-
1. ॠणांक अंक का धनात्मक मान लेते हैं।
2. इस मान का परम मित्र लिखते हैं।
3. ॠणांक अंक के पूर्व अंक पर एकन्यून चिन्ह लगाते हैं।
Example:
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संकेत 1. दहाई सथान के ॠणात्मक अंक से प्रारम्भ करते हैं। 2. के धनात्मक मान 3 का परम मित्र अंक 7 लिखकर इसके पूर्व के अंक 4 पर एक न्यून चिन्ह लगाते हैं। 3. के धनात्मक मान 2 का परम मित्र अंक 8 लिखकर इसके पूर्व के अंक 7 पर एक न्यून चिन्ह लगाते हैं। |
निखिलम् विधि द्वारा ॠणांक (विनकुलम्) संख्या को सामान्य संख्या में बदलना-
1. ॠणांक के ऊपर शिरो रेखा की चिन्ता नही करते हुए चरमं अंक को 10 में से घटाते हैं।
2. शेष निखिलम् अंकों को 9 में से घटाते हैं।
3. शेषफल से पूर्व अंक पर एक न्यून चिन्ह लगाते हैं।
Example:
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संकेत 1. ॠणांक के ऊपर शिरो रेखा की चिन्ता नही करते हुए चरमं अंक को 10 में से घटाते हैं, 10-2=8 2. निखिलम् अंक को 9 में से घटाते हैं, 9-3=6 3. 4 पर एक न्यून चिन्ह लगाते हैं। |
It is interesting to see that the conversions can be shown in many different ways.
Example: 87 को निम्न प्रकार से व्यक्त किया जा सकता हैं।
A. Addition and subtraction using vinkulam
Adding a bar digit or Rekhank to a digit means the digit is subtracted.
किसी रेखांक को किसी धनात्मक अंक में जोड़ने का मतलब घटाना होता हैं।
Example:
3 + 2' = 1
5 + 2' = 3
4 + 4' = 0
B. Subtracting a bar digit or Rekhank to a digit means the digit is added.
किसी रेखांक को किसी धनात्मक अंक में से घटाने का मतलब जोड़ना होता हैं।
Example:
3 – 2' = 5
2 – 2' = 4
6 – 2' = 8
3 – 3' = 6
छोटे होते हैं।
घटाते समय इनका नाम वियोजक होता है।
Reducible Digits
Digits which
Exercise (R –> L)
7 8 3
– 4 5 8
3 3 5'. Or
3 2 5 Ans.
6 8 2
– 4 5 8
2 3 6'. Or
2 2 4 Ans.
6 8 2 1
– 4 5 8 3
2 3 6' 2' Or
2 2 3 8 Ans.
C. Multiplication using rekhank
* Product of two positive digits or two negative (Rekhanks) digits is always positive.
दो धनात्मक या ॠणात्मक संख्याओं का गुणनफल हमेशा धनात्मक संख्या होती हैं।
Example:
3' × 3' = 9
3 × 3 = 9
2' × 3' = 6
2 × 3 = 6
3' × 4' = 12
3 × 4 = 12
* Product of one positive digit and one Rekhank is always negative or Rekhank.
एक धनात्मक तथा एक रेखांक संख्या का गुणनफल हमेशा ॠणात्मक या रेखांक संख्या प्राप्त होती हैं।
Example:
3 × 3' = 9'
3' × 3 = 9'
2' × 3 = 6'
2 × 3' = 6'
3 × 4' = 12'
3' × 4 = 12'
D. Division using rekhank
Division of one positive by another positive or division of one Rekhank by another Rekhank is always positive.
किसी धनात्मक संख्या को धनात्मक संख्या से या किसी ॠणात्मक संख्या को ॠणात्मक संख्या से भाग करने पर हमेशा धनात्मक संख्या होती हैं।
Example:
3' ÷ 3' = 1
6 ÷ 3 = 2
12' ÷ 3' = 4
12 ÷ 4 = 3
21' ÷ 3' = 7
32 ÷ 4 = 8
Division of a positive by a Rekhank or vice versa or inverse is Rekhank or negative.
किसी धनात्मक संख्या को ॠणात्मक संख्या से या किसी ॠणात्मक संख्या को धनात्मक संख्या से भाग करने पर हमेशा ॠणात्मक संख्या या रेखांक संख्या प्राप्त होती हैं।
Example:
3' ÷ 3 = 1'
6 ÷ 3' = 2'
12' ÷ 3 = 4'
12 ÷ 4' = 3'
21' ÷ 3 = 7'
32 ÷ 4' = 8'
2. Vinculum Numbers:
Vinculum means bar(line) present over the symbol/digit.
Sanskrit Name:
विनक्ल्म्
English Translation:
Complement of a number.
Vinculum Process or Vinculum Numbers are the very basics of Vedic Mathematics.
Vinculum Numbers is concept used in Vedic Mathematics and are those numbers which have atleast 1 digit which is negative (having bar over them). Also called as Bar Numbers.
As seen earlier Normal Number can be written as
2345 = 2000 + 300 + 40 + 5
Similarly Vinculum Numbers can be written as and can be converted to normal numbers as below:
- complement with a Bar over it.
- (a).If next digit is >= 5, take its 9’s
complement with a Bar over it & continue this till a digit < 5 is
obtained.
(b). Increment <5 digit by 1. - Continue (1) and (2) till complete number is covered.
Subtraction using Vedic Mathematics:
Most of us have difficulty and uncertainty whenever we need to do subtraction, especially when it comes to subtraction of larger digits from smaller digits.
Vedic Mathematics’s Vinculum process can also be used for Subtracting 2 numbers. (How ?? watch my below Video “Vedic Mathematics -1 (Vinculum 1of3)”).
Process (R –> L):
- If bigger digit is to be subtracted from smaller digit, then just perform Bigger digit – Smaller digit and write bar over it.
- If smaller digit is to be subtracted from bigger digit then carry usual method.
- In final answer (Vinculum number) convert to normal number using Vinculum process.
Multiplication and Division examples in Vedic Maths which uses Vinculum Process will be seen in respective topics.
3. Quotient and Remainders:
This is another important concept of Vedic Mathematics. We will follow the below formula and the concept that Remainder is ALWAYS < Divisor.
Dividend = Quotient x Divisor + Remainder
Examples:
33 ÷ 6 = 5/3 … where 5 -> Quotient & 3
-> Remainder
34 ÷ 6 = 5/4 … where 5 -> Quotient & 4 -> Remainder
35 ÷ 6 = 5/5 … where 5 -> Quotient & 5 -> Remainder
36 ÷ 6 = 6/0 … where 6 -> Quotient & 0 -> Remainder
So on observation we can say that the Remainder can never be >= Divisor.
But in Vedic mathematics in some examples depending on some criteria we need to play with Quotients and Remainders i.e. use Remainder >= Divisor for carrying out the process. But the in final answer Remainder can never be >= Divisor.
Thus for carrying out the process above examples (from bottom to top) can also be written as: ( ( watch my below Video “Vedic Mathematics -4 (Playing with Quotients and Remainders)”).)
36 ÷ 6 = 5/6 OR 4/12 OR 3/18 and so on.
35 ÷ 6 = 4/11 OR 3/17 and so on.
And vice versa for calculating final answer(Remainder can never be >= Divisor).
If we obtained Remainder(R) which is >=
Divisor(D), we divide R by D and corresponding obtained quotient is added with
obtained Q and new remainder becomes our R.
Lets say we obtained Q=4 & R=12, and D used was 11. As R > D, Apply
above rule.
So Actuals are Q= 5 and R=1. Q=4 & R=18, and D used was 6. As R > D,
Apply above rule. So Actuals are Q= 7 and R=0.aring is caring!
Basics of Vedic Mathematics:
- Place Value System
- Vinculum Numbers (English Meaning: Complement of a Number).
- Work with Quotients & Remainders.
विनजीत वैदिक अंकगणित पुस्तक || 1 || अध्याय 08.03.1 || विनकुलम् अंक
साभार वैदिक गणित
(1) साधारण एवं वैदिक संख्याएं :
गणित की साधारण प्रणाली में हम जिन संख्याओं का प्रयोग करते हैं उनके सभी अंक धनात्मक होते हैं। उदाहरणार्थ: संख्या 123546 के सभी अंक 1, 2, 3, 5, 4 और 6 धनात्मक हैं।
परन्तु वैदिक गणित में ऐसी संख्याओं का भी प्रयोग किया जाता है जिनके सभी अंक या तो धनात्मक हों या ऋणात्मक हों या फिर दोनों ही प्रकार के अर्थात धनात्मक तथा ऋणात्मक दोनों ही प्रकार के हो सकते हैं।
अब प्रश्न उठता है कि वैदिक गणित में ऋणात्मक अंकों का प्रयोग क्यों और किस प्रकार से होता है?
इस प्रश्न का उत्तर है कि वैदिक गणित में प्रयुक्त संख्याओं के सभी अंक 5 अथवा 5 से छोटे रखे जाते हैं। वैदिक गणित में बड़े अंकों 6, 7, 8, 9 का प्रयोग नहीं किया जाता है। वैदिक गणित इन अंको की कमी कैसे पूरी करता है?
पांच से बड़े अंकों 6, 7, 8, 9 को छोटे अंकों में परिवर्तित कर लेते हैं। पांच से बड़े अंकों 6, 7, 8, 9 को छोटे अंकों में परिवर्तित करने की प्रक्रिया में ऋणात्मक अंक प्राप्त होते हैं। इन ऋणात्मक अंकों को विनकुलम् अंक कहते हैं ।
अतः वैदिक संख्या में धनात्मक तथा ऋणात्मक दोनों ही प्रकार के अंकों का समावेश हो जाता है।
(1) Ordinary and Vedic numbers:
Generally, all the digits of the numbers we use in mathematics are positive. For example: All the digits 1, 2, 3, 5, 4 and 6 of the number 123546 are positive.
But in Vedic mathematics such numbers are also used whose all digits are either positive or negative or can be of both types i.e. both positive and negative.
Now the question arises that why and how are negative numbers used in Vedic mathematics?
The answer to this question is that all the digits of the numbers used in Vedic mathematics are kept 5 or smaller than 5. Big numbers 6, 7, 8, 9 are not used in Vedic mathematics. How does Vedic mathematics compensate for the lack of these numbers?
Digits greater than five, 6, 7, 8, 9 are converted into smaller digits. In the process of converting numbers larger than five, 6, 7, 8, 9 into smaller numbers, negative numbers are obtained. These negative numbers are called Vinkulam numbers.
Therefore, Vedic numbers include both positive and negative numbers.
(2) विनकुलम् :
वैदिक गणित में प्रयुक्त संख्याओं को उनके अंकों को ऋणात्मक रूप में लिखने की प्रक्रिया को विनकुलम् कहते हैं ।
अतः वैदिक गणित में ऋणात्मक अंक (–4) को, अंक 4 के ऊपर ऋणात्मक चिन्ह या डेश लगाकर 4' व्यक्त किया जाता है जो (–4) का विनकुलम् 4' अथवा बिनकुलम् कहलाता है।
(2) Vinakulam:
The process of writing the numbers in negative form used in Vedic mathematics is called Vinakulam.
Therefore, in Vedic mathematics, the negative number (–4) is expressed as 4' by putting a negative sign or dash above the number 4, which is called Vinakulam 4' or Binakulam of (–4).
(3) विनकुलम् का सिद्धान्त :
पांच से बड़े अंकों 6, 7, 8, 9 को छोटे अंकों में परिवर्तित करने की प्रक्रिया को विनकुलम् कहते हैं । जिससे इनके छोटे व ऋणात्मक अंक प्राप्त होते हैं । इन छोटे व ऋणात्मक अंकों को विनकुलम् अंक कहते हैं ।
जैसा कि हम पहले ही बता चुके हैं कि वैदिक गणित में प्रयुक्त होने वाली संख्याओं में सभी अंक 5 अथवा 5 से छोटे रखे जाते हैं। अतः संख्या मैं जो भी अंक 5 से बड़े (6, 7, 8, 9) होते हैं, उन सभी अंकों के स्थान पर उनके विनकुलम् अंक रख कर उपयुक्त क्रिया करते हैं।
(3) Principle of Vinkulam:
The process of converting numbers 6, 7, 8, 9 larger than five into smaller numbers is called Vinkulam. Due to which they get small and negative marks. These small and negative numbers are called Vinkulam numbers.
As we have already told that in the numbers used in Vedic mathematics, all the digits are kept 5 or smaller than 5. Therefore, all the digits in the number which are greater than 5 (6, 7, 8, 9) are replaced by their Vinkulam digits and appropriate actions are performed.
(4) किसी अंक का विनकुलम् अंक या विनकुलम् ज्ञात करना :
5 से बड़े जिस अंक का भी विनकुलम् ज्ञात करना होता है। विनकुलम् बड़े अंक का '10 से ऋणात्मक विचलन' होता हैं।
चूंकि 5 से बड़े अंक 6, 7, 8 और 9 सभी 10 से कम हैं, इसलिए इनके 10 से विचलन ऋणात्मक चिन्ह वाले होंगे। विचलन से प्राप्त इन अंकों को उनके सामने ऋणात्मक चिन्ह (–) या ऊपर डेश चिन्ह ( ' ) लगाकर प्रदर्शित किया जाता है। इन रेखांकित अंकों को विनकुलम् या विनकुलम् अंक कहते हैं।
उदाहरणार्थ
अंक 6 का विनकुलम् ज्ञात करना।
अंक 6 का 10 से विचलन = 6 –10 = –4 या 4'
अंक 6 का विनकुलम् (अंक) = –4 या 4'
इसी प्रकार, अंक 7 का विनकुलम् (अंक) = 7–10=–3 या 3'
तथा
अंक 8 को विनकूलम् (अंक) = 8–10=–2 या 2'
अंक 9 का विलकुलम् (अंक) =9–10=–1 या 1'
(4) To find Vinkulam number or Vinkulam of a number:
Vinkulam of any number greater than 5 has to be found. Vinkulam is the 'negative deviation from 10' of the larger number.
Since numbers greater than 5, 6, 7, 8 and 9 are all less than 10, their deviations from 10 will have negative signs. These numbers obtained from deviations are displayed by putting a negative sign (–) in front of them or a dash sign (') above them. These underlined numbers are called Vinkulam or Vinkulam numbers.
For Example
To find Vinkulam of number 6.
Deviation of number 6 from 10 = 6 –10 = –4 or 4'
Vinkulam (digit) of number 6 = –4 or 4'
Similarly, Vinkulam (digit) of number 7 = 7–10=–3 or 3'
And
Vinkulam (digit) of number 8 = 8–10=–2 or 2'
Vilakulam (digit) of number 9 =9–10=–1 or 1'
(5) विनकुलम् संख्याएँ :
धनात्मक तथा ऋणात्मक दोनों प्रकार के अंकों से बनी संख्याएँ, विनकुलम् सख्याएँ कहलाती है।
याद रखिये कि विनकुलम् संख्याओं में 5 अथवा 5 से छोटे अंकों का हो प्रयोग होता है। इन वैदिक संख्याओं में 5 से बड़े अंक उनके विनकुलम् अंकों के रूप में प्रयुक्त होते हैं।
(5) Vinkulam numbers:
Numbers made up of both positive and negative digits are called Vinkulam numbers.
Remember that in Vinkulam numbers, digits 5 or less than 5 are used. In these Vedic numbers, digits greater than 5 are used as their Vinkulam digits.
(6) निखिलम् सूत्र :
निखिलम् सूत्र है –
'निखिलम् नवत: चरमं दशतः' अर्थात 'प्रत्येक अंक को 9 में से तथा अन्तिम दाएँ अंक को 10 में से घटाओ।'
निखिलम् सूत्र की सहायता से अंकों के समूह का भी विनकुलम् सरलतापूर्वक प्राप्त कर सकते है। अतः यह निखिलम् सूत्र दी हुई सामान्य संख्या को विनकुलम् संख्या के रूप में लिखने का एक सशक्त साधन है।
(6) Nikhilam Sutra:
Nikhilam Sutra is –
'Nikhilam Navatah Charam Dashatah' means 'Subtract each digit from 9 and the last right digit from 10.'
With the help of Nikhilam Sutra, Vinakulam of a group of numbers can be easily obtained. Hence, this Nikhilam formula is a powerful means of writing the given ordinary number in the form of Vinkulam Number.
(7) साधारण संख्या को विनकूलम् संख्या के रूप में लिखना:
किसी संख्या को विनकुलम् रूप में लिखने के लिए उस संख्या में आने वाले 5 से बड़े अंकों को उनके विनफुलम् अंकों से बदल दिया जाता है। और संख्या में जो अंक 5 से छोटे होते हैं उन्हें ज्यों का त्यों ही लिख दिया जाता है।
अब हम विनकुलम् अंक लिखने की पूरी विधि का विस्तार से वर्णन करेंगे।
(7) Writing an ordinary number in the form of Vinkulam number:
To write a number in Vinkulum form, the digits greater than 5 in that number are replaced with their Vinkulum digits. And the digits in the number which are less than 5 are written as they are.
Now we will describe in detail the complete method of writing Vinkulam numbers.
(a) जब संख्या में 'केवल एक' अंक का विनकुलम् ज्ञात करना हो :
माना हमें संख्या 382 को विनकुलम् रूप में लिखना है। यहाँ केवल अंक 8 का विनकुलम् अंक ज्ञात करना होगा।
(a) When the vinculum of 'only one' digit in the number is to be found:
Suppose we have to write the number 382 in Vinkulam form. Here only Vinkulam digit of number 8 has to be found.
Working Method
कार्य विधि
(i) सर्वप्रथम अंक 8 का 10 से विचलन शात करो जो (–2) है। अंक 8 का विनकुलम् (अंक) = 2'
(ii) अंक 8 के बायीं ओर स्थित अंक 3 में 1 की वृद्धि करो। अतः 3+1=4
(iii) दी हुई संख्या 382 में, 8 के स्थान पर 2 तथा बायीं ओर स्थित अंक 3 के स्थान पर 4 लिखने पर, 382–42'2
अतः संख्या 382 का विनकुलम् रूप 42'2 है 1 संक्षेप में उक्त क्रिया निम्न प्रकार की जा सकती है--
382=(•3)(8–10)2=4(–2)2=42'2
(i) First find the deviation of the number 8 from 10 which is (–2). Vinkulam (digit) of digit 8 = 2'
(ii) Increase the digit 3 on the left of the digit 8 by 1. Hence 3+1=4
(iii) In the given number 382, on writing 2 in place of 8 and 4 in place of 3 on the left, 382–42'2
Hence, Vinkulam form of number 382 is 42'2. In short, the above action can be done in the following way-
382=(•3)(8–10)2=4(–2)2=42'2
ऊपर दी गयी विधि की व्याख्या :
(i) 5 से बड़े अंक का 10 से विचलन लेने पर जो अंक मिलता है उसके ऊपर ऋणात्मक चिन्ह (–) या डेश चिन्ह ( ' ) लगाओ। यह उस अंक का विनकुलम् होगा। इस अंक के स्थान पर उसका विनकुलम् (अंक) लिख दो ।
(ii) जिस अंक का विनकुलम् प्राप्त किया है उसके ठीक बाईं और के अंक को 1 बढ़ाकर लिखो।
Explanation Of The Above Method:
(i) Put a negative sign (–) or a dash sign (') on the number obtained by taking the deviation of a number greater than 5 from 10. This will be the vinakulam of that number. In place of this number, write its Vinakulam (number).
(ii) Write the digit immediately to the left of the number for which Vinkulam has been obtained by increasing it by 1.
आवश्यक नोट -
यदि विनकुलम् ज्ञात करने वाले अंक के बायीं ओर कोई भी अंक न हो तो वहाँ शून्य लिख दिया जाता है । अर्थात 0+1=1
Important Notes -
If there is no digit on the left side of the number used to find Vinkulam, then zero is written there. i.e. 0+1=1
उदाहरणार्थ
संख्या 94 या 094 को विनकुलम् रूप में लिखिए।
अंक 9 का विनकुलम् (अंक) = •0(9–10)4
अंक 9 के बायीं ओर 0 में 1 जोड़ने पर,
0+1=1
अतः
94=11'4
संक्षेप में
94 = 094 = (0+1)(9–10)4=1(–1)4=11'4
For example
Write the number 94 or 094 in Vinkulam form.
Vinkulam (digit) of number 9 = •0(9–10)4
By adding 1 to 0 on the left side of the number 9,
0+1=1
Therefore
94=11'4
in short
94 = 094 = (0+1)(9–10)4=1(–1)4=11'4
(b) जब संख्या में 'एक से अधिक' अंकों का विनकुलम् ज्ञात करना हो :
यदि दी हुई संख्या में 5 से बड़े अंक कई स्थानों पर आ रहे हों तो सर्वप्रथम उन अंकों के भिन्न-भिन्न समूह बना लेते हैं। यदि संख्या में दो या दो से अधिक ऐसे अंक हों तो उन सभी अंकों को अलग अलग समूह में रखा जाता है ।
(b) When the Vinkulam of 'more than one' digits in the number is to be found:
If digits greater than 5 are appearing at many places in a given number, then first of all make different groups of those digits. If there are two or more such digits in the number, then all those digits are kept in separate groups.
अब विनकुलम् इस प्रकार ज्ञात किया जाता है।
(i) प्रत्येक समूह में जो अंक सबसे दायीं ओर हो उसे 10 से घटाकर तथा शेष अंकों को क्रमवार 9 से घंटाकर जो अंक मिलते हैं उनके ऊपर ऋणात्मक चिन्ह (– ) या डेश चिन्ह ( ' ) लगाकर सभी अंकों के विनकुलम् बना लेते हैं। अब समूह के प्रत्येक अंक को उसके विनकुलम् (अंक) से बदल देते हैं ।'
(ii) प्रत्येक समूह के ठीक बाईं ओर स्थित अंक के मान में एक अंक की वृद्धि कर देते हैं अर्थात् वहाँ के अंक को 1 बढ़ाकर लिख देते हैं।
Now Vinakulam is known as follows.
(i) In each group, by subtracting the rightmost digit from 10 and multiplying the remaining digits by 9, we make Vinakulam of all the digits by putting negative sign (–) or dash sign (') on the digits obtained. Now let us replace each digit of the group with its vinakulam (digit).
(ii) The value of the digit located immediately to the left of each group is increased by one digit, that is, the digit there is written by increasing it by 1.
आवश्यक नोट--
यदि किसी समूह में केवल एक अंक हो तो उसे 10 से घटाकर उस अंक का विनकुलम् ज्ञात करके वहाँ लिख देते हैं।
उदाहरण -1.
संख्या 782893 को विनकुलम् रूप में लिखिए ।
दी हुई संख्या= 782893 = 0782893 =0(78)2(89)3
'यहाँ 5 से बड़े अंकों के दो समूह (78) और (89) बनते हैं।
समूह (89) का विनकुलम् ज्ञात करना :
निखिलम् सूत्र से, दायीं ओर के अंक 9 का 10 से विचलन लेने पर, अंक 9 का विनकुलम् = –1 तथा दूसरे अंक 8 का 9 से विचलन लेने पर, अंक 8 का विनकुलम् = –1
समूह (78) का विनकुलम् ज्ञात करना :
निखिलम् सूत्र से, दायीं ओर के अंक 8 का 10 से विचलन लेने पर, अंक 8 का बिनकुलम् = –2 तथा दूसरे अंक 7 का 9 से विचलन लेने पर, अंक 7 का विनकुलम् –2
अब प्रत्येक समूह में आने वाले अंकों के स्थान पर उनके विनकुलम् अंक रख देंगे तथा प्रत्येक समूह के बायीं ओर स्थित अंक के मान में एक की वृद्धि कर देंगे। अतः विनकुलम् रूप में,
0782893 = 12'2'31'1'3
ध्यान दीजिए कि यहाँ समूह (78) के बायीं ओर कोई अंक नहीं है। अतः बायीं ओर शून्य अंक मान कर उसमें एक की वृद्धि करके 0+1=1 लिखा गया है।
Important note--
If there is only one digit in a group, then by subtracting it from 10, we find the Vinkulam of that digit and write it there.
Example 1.
Write the number 782893 in Vinkulam form.
Given number = 782893 = 0782893 =0(78)2(89)3
'Here two groups of numbers greater than 5 are formed (78) and (89).
To find the vinculum of group (89):
From Nikhilam formula, taking the deviation of the rightmost digit 9 from 10, Vinkulam of digit 9 = –1 and taking the deviation of the second digit 8 from 9, Vinkulam of digit 8 = –1
To find the vinculum of group (78):
From Nikhilam Sutra, taking the deviation of the rightmost digit 8 from 10, the binakulam of the digit 8 = –2 and taking the deviation of the second digit 7 from 9, the binakulam of the digit 7 is –2.
Now in place of the digits appearing in each group, their Vinkulam digits will be placed and the value of the digit on the left side of each group will be increased by one. Hence in Vinakulam form,
0782893 = 12'2'31'1'3
Notice that there is no number on the left side of the group (78). Therefore, considering the number zero on the left side and increasing it by one, it is written as 0+1=1.
संक्षेप में उक्त गणना इस प्रकार की जा सकती है-
In short the above calculation can be done as follows
782893= 0782893 =0(78)2(89)3
=(•0)[(7–9)(8–10)](•2)[(8–9)(9–10)3
=1[(–2)(–2)]3[(–1)(–1)]3
= 12'2'31'1'3
उदाहरण-2.
संख्या-1918176 को विनकुलम् रूप में लिखिए।
दी हुई संख्या = 1918176=1(9)1(8)1(76)
यहाँ 5 से बड़े अंकों के तीन समूह (9), (8) और (76) बनते हैं।
समूह (76) का विनकुलम् ज्ञात करना :
दायीं ओर' के अंक 6 का 10 से तथा दूसरे अंक 7 का 9 से विचलन लेने पर
अंक 6 का बिनकुलम् = –4 या 4'
अंक 7 का विनकुलम् –2 या 2'
समूह (8) का विनकुलम् ज्ञात करना :
अंक 8 का 10 से विचलन लेने पर,
अंक 8 का वितकुलम् = –2 या 2'
समूह (9) का विनकुलम् ज्ञात करना :
अंक 9 का 10 से विचलन लेने पर,
अंक 9 का विनकुलम् = –1 या 1'
अब प्रत्येक समूह में आने वाले अंकों के स्थान पर उनके वितकुलम् अंक रख देंगे तथा प्रत्येक समूह के बायीं ओर स्थित अंक के मान में एक की बुद्धि कर देंगे।
अतः विनकुलम् रूप में,
1918176 = 21'22'22'4'
Example 2.
Write the number-1918176 in Vinkulam form.
Given number = 1918176=1(9)1(8)1(76)
Here three groups of numbers greater than 5 are formed (9), (8) and (76).
To find the vinculum of group (76):
By taking the deviation of the right side digit 6 by 10 and the deviation of the second digit 7 by 9.
Binakulam of number 6 = –4 or 4'
Vinkulam of number 7 –2 or 2'
To find Vinkulam of group (8):
Taking deviation of number 8 from 10,
Vitakulam of number 8 = –2 or 2'
To find Vinkulam of group (9):
Taking the deviation of the number 9 from 10,
Vinkulam of number 9 = –1 or 1'
Now in place of the digits appearing in each group, their Vitakulam digits will be placed and the value of the digit located on the left side of each group will be increased by one.
Hence in Vinakulam form,
1918176 = 21'22'22'4'
संक्षेप में उक्त गणना इस प्रकार की जा सकती है
In short the above calculation can be done as follows-
1918176 = 1(9)1(8)1(76)
=(•1) (9-10)(•1)(8-10)(•1) [(7-9)(6-10)]
=2(–1)2(–2)2[(–2)(–4)]
=21' 2 2' 2 2' 4'
भाग 02
Part 02
विविनकुलम् संख्या को सामान्य (वास्तविक) रूप में लिखना :
विनकुलम् संख्या को सामान्य संख्या में बदलने की क्रिया को विविनकुलम् कहलाता है। यह विनकुलम् संख्या ज्ञात करने की विलोम क्रिया है।
अतः दी हुई विनकुलम् संख्या के सभी रेखांकित (ऋणात्मक) अंकों को धनात्मक अंकों में अर्थात् मूल अंकों में परिवर्तित करके उस संख्या को वास्तविक रूप में लिखा जाता है ।
Writing Vivinkulam numbers in normal (real) form:
The process of converting Vinkulam numbers into normal numbers is called Vivinkulam. This is the inverse of finding Vinkulam number.
Therefore, by converting all the underlined (negative) digits of the given Vinkulam number into positive digits i.e. into original digits, that number is written in its real form.
आइए अब हम पूरी विधि का विस्तार से वर्णन करेंगे।
(a) जब केवल एक विनकुलम् अंक को मूल अंक में बदलना हो :
यहाँ निम्नलिखित विधि अपनायी जाती है-
(i) संख्या में जो भी विनकुलम् अंक (रेखांकित अंक) या डेश अंक हो उसे 10 में से घटाकर उसका धनात्मक पूरक अंक ज्ञात कर लेते हैं। अब इस विनकुलम् अंक के स्थान पर इस पूरक अंक की लिख देते हैं ।
(ii) विनकुलम् अंक (रेखांकित अंक) या डेश अंक के ठीक बायीं ओर के अंक के मान में एक की कमी करके लिख देते हैं।
Let us now describe the entire method in detail.
(a) When only one Vinkulam digit is to be converted into the original digit:
Here the following method is adopted-
(i) Whatever Vinkulam digit (underlined digit) or dash digit is there in the number, its positive complement digit is found by subtracting it from 10. Now in place of this Vinakulam number, let us write this supplementary number.
(ii) The value of the digit immediately to the left of Vinkulam digit (underlined digit) or dash digit is written by decreasing it by one.
उदाहरण -1.
★ अंक 3' को सामान्य अंक में बदलो
पूरक अंक (10–3) = 7
★ संख्या 27'1 को वास्तविक (सामान्य) रूप में लिखिए ।
अंक 7' का पूरक अंक(10–7) = *2(10–7)1
अंक 7 के बायीं ओर स्थित अंक 2 में से 1 कम करने पर, 2–1=1 अतः दी हुई संख्या 27'1 में, अंक 7 के स्थान पर 3 तथा बायीं ओर के अंक 2 के स्थान पर 1 लिखने पर,
27'1=131
Example 1.
★ Convert digit '3' to normal digit
Complementary digits (10–3) = 7
★ Write the number 27'1 in real (normal) form.
Complement of digit 7'(10–7) = *2(10–7)1
By subtracting 1 from the digit 2 situated on the left side of the number 7, 2–1=1 Hence, in the given number 27'1, on writing 3 in place of the digit 7 and 1 in place of the digit 2 on the left side,
27'1=131
संक्षेप में
In Short
27'1=(*2)(10–7)1 =131
उदाहरण - 2.
संख्या 8' 3 4 को वास्तविक रूप में लिखिए।
यहाँ अंक 8' के स्थान पर इसका पूरक अंक 10 – 8 = 2 लिखा जाएगा तथा 8 के बायीं ओर शून्य मानकर वहाँ 0–1=–1 = 1' लिखा जायेगा।
8'34 = 08'34 = *0(10–8)34 = *1(234)
= –1000+234 = –766
Example - 2.
Write the number 8' 3 4 in real form.
Here, in place of the number 8', its complementary number 10 – 8 = 2 will be written and considering zero on the left side of 8, it will be written as 0–1=–1 = 1'.
संक्षेप में
In Short
8'34 = 08'34 = *0(10–8)34 = *1(234)
= –1000+234 = –766
(b) संख्या के एक से अधिक विनकुलम् (रेखांकित) या डेश अंकों को मूल रूप में परिवर्तित करना :
यदि दी हुई संख्या में विनकुलम् (रेखांकित) या डेश अंक अलग अलग कई स्थानों पर आ रहे हैं तो सर्वप्रथम उन विनकुलम् (रेखांकित) या डेश अंकों के भिन्न-भिन्न समूह बना लेते हैं। यदि दो या दो से अधिक विनकुलम् (रेखांकित) या डेश अंक हों तो वे सभी अंक एक ही समूह में रखे जायेंगे ।
अब निखिलम् सूत्र की सहायता से, प्रत्येक समूह में, सबसे दायीं ओर के अंक को 10 में से घटाकर उसका धनात्मक पूरक अंक ज्ञात कर लेते हैं।
समूह के शेष अंकों में से प्रत्येक अंक को 9 में से घटाकर उनके पूरक अंक ज्ञात कर लेते हैं।
अन्त में, प्रत्येक समूह के सभी विनकुलम् अंकों को उनके पूरक अंकों से बदल देते हैं । अब प्रत्येक समूह के ठीक बायीं ओर स्थित अंक के मान में एक कम कर देते हैं।
नोट
यदि किसी समूह में केवल एक विनकुलम् अंक हो तो उसे 10 में से घटाकर उसका पूरक अंक ज्ञात करते हैं।
(b) Converting more than one Vinakulam (underlined) or dash digit of the number to its original form:
If Vinkulam (underlined) or dash digits are appearing at different places in a given number, then first of all make different groups of those Vinkulam (underlined) or dash digits. If there are two or more Vinkulam digits then all those digits will be kept in the same group.
Now with the help of Nikhilam Sutra, in each group, we find its positive complement digit by subtracting the rightmost digit from 10.
The remaining digits of the group are found by subtracting each digit from 9 to find their complementary digits.
Finally, all the Vinkulam digits of each group are replaced by their complementary digits. Now the value of the digit situated on the immediate left of each group is reduced by one.
Note: If there is only one Vinkulam digit in a group, then its complementary digit is found by subtracting it from 10.
उदाहरण - 1.
संख्या 33'44 को वास्तविक रूप में लिखिए ।. दी हुई संख्या =3 3'4 4'
=3(3')4(4') अंक हैं। यहाँ रेखांकित अंकों के दो समूह ( 3' ) और (4') बनते हैं जिनमें एक-एक समूह ( 4' ) के अंक 4 का पूरक अंक 10–4=6 तथा इस समूह के बायीं ओर स्थित अंक 4 में से 1 कम करने पर, 4–1= 3 पुनः समूह ( 3' ) के अंक 3 का पूरक अंक=10–3=7 तथा समूह के बायीं ओर स्थित अंक 3 में से 1 कम करने पर, अतः संख्या 3 3' 4 4' का वास्तविक रूप निम्न है- 2736
Example 1.
Write the number 33'44 in real form. Given number =3 3'4 4'
=3(3')4(4') numbers. Here two groups of underlined digits (3') and (4') are formed in which the complement of digit 4 of each group (4') is 10–4=6 and the digit situated on the left side of this group is 1 less than 4. On doing this, 4–1= 3 Again, the complement of number 3 of the group (3') = 10–3 = 7 and on subtracting 1 from the number 3 on the left side of the group, hence the number 3 3' 4 4' The actual form is as follows- 2736
संक्षेप में
In Short
33'44'
= *3 (10–3) *4 (10–4)
= 2 (7) 3 (6)
=2736
उदाहरण-2
संख्या 26'3'42'35'1' को वास्तविक रूप में लिखिए :
दी हुई संख्या = 26'3'42'35'1= 2(6'3')4(2')3(5'1')
यहाँ रेखांकित अंकों के तीन समूह (6' 3' ) , (2') और (5' 1') बनते हैं।
निखिलम् सूत्र से, प्रत्येक समूह में, दायीं ओर के अंक को 10 में से तथा शेष बैंकों को 9 में से घटाया जायेगा। अन्त में, प्रत्येक समूह के बायीं और स्थितअंक में 1 की कमी की जायेगी ।
Example-2
Write the number 26'3'42'35'1' in real form:
Given number = 26'3'42'35'1= 2(6'3')4(2')3(5'1')
Here three groups of underlined digits are formed (6' 3' ), (2') and (5' 1').
Using the Nikhilam formula, in each group, the number on the right will be subtracted from 10 and the remaining banks will be subtracted from 9. Finally, the leftmost points of each group will be reduced by 1.
= 2(6'3')4(2')3(5'1')
= *2 [(9 –6)(10 – 3)] *4 (10– 2) *3 [(9 – 5)(10 – 1)]
=1 37 3 8 2 49
This is another important concept of Vedic Mathematics. We will follow the below formula and the concept that Remainder is ALWAYS < Divisor.
3. Quotient and Remainders:
This is another important concept of Vedic Mathematics. We will follow the below formula and the concept that Remainder is ALWAYS < Divisor.
Dividend = Quotient x Divisor + Remainder
Examples:
33 ÷ 6 = 5/3 … where 5 -> Quotient & 3 ->
Remainder
34 ÷ 6 = 5/4 … where 5 -> Quotient & 4 -> Remainder
35 ÷ 6 = 5/5 … where 5 -> Quotient & 5 -> Remainder
36 ÷ 6 = 6/0 … where 6 -> Quotient & 0 -> Remainder
So on observation we can say that the Remainder can never be >= Divisor.
But in Vedic mathematics in some examples depending on some criteria we need to play with Quotients and Remainders i.e. use Remainder >= Divisor for carrying out the process. But the in final answer Remainder can never be >= Divisor.
Thus for carrying out the process above examples (from bottom to top) can also be written as: ( ( watch my below Video “Vedic Mathematics -4 (Playing with Quotients and Remainders)”).)
36 ÷ 6 = 5/6 OR 4/12 OR 3/18 and so on.
35 ÷ 6 = 4/11 OR 3/17 and so on.
And vice versa for calculating final answer(Remainder can never be >= Divisor).
If we obtained Remainder(R) which is >= Divisor(D), we
divide R by D and corresponding obtained quotient is added with obtained Q and
new remainder becomes our R.
Lets say we obtained Q=4 & R=12, and D used was 11. As R > D, Apply
above rule.
So Actuals are Q= 5 and R=1. Q=4 & R=18, and D used was 6. As R > D,
Apply above rule. So Actuals are Q= 7 and R=0.aring is caring!
Basics of Vedic Mathematics:
- Place Value System
- Vinculum Numbers (English Meaning: Complement of a Number).
- Work with Quotients & Remainders.
1. Place Value System:
It denotes the value present at particular place. Place Value concept is used for Vinculum Numbers (For conversion of Vinculum Numbers to normal numbers and vice versa).
Example: 2345
- 5 is present at Units place. Hence Place Value of 5 is 5.
- 4 is present at Tens place. Hence Place Value of 4 is 40.
- 3 is present at Hundreds place. Hence Place Value of 3 is 300.
- 2 is present at Thousands place. Hence Place Value of 2 is 2000.
Hence
2345 = 2000 + 300 + 40 + 5
2. Vinculum Numbers:
Vinculum means bar(line) present over the symbol/digit.
Sanskrit Name:
विनक्ल्म्
English Translation:
Complement of a number.
Vinculum Process or Vinculum Numbers are the very basics of Vedic Mathematics.
Vinculum Numbers is concept used in Vedic Mathematics and are those numbers which have atleast 1 digit which is negative (having bar over them). Also called as Bar Numbers.
As seen earlier Normal Number can be written as
2345 = 2000 + 300 + 40 + 5
Similarly Vinculum Numbers can be written as and can be converted to normal numbers as below:
Another Approach for converting Vinculum Number to General Number, I generally remember this from R –> L as below (for better approach watch my below Video “Vedic Mathematics -1 (Vinculum 1of3)”).
- Convert 1st Bar digit from Right side to Normal digit (By taking its 10’s complement)
- Decrement the previous digit by 1 (If it comes negative then repeat these 2 steps.)
Vinculum (Bar Number) –> Normal Number (Check the video)
R –> L
- Find 1st Bar digit, take its 10’s complement.
- (a).If next digit is again a Bar digit, take its 9’s
complement continue (a) until a non-bar digit is obtained.
(b). Decrement non-bar digit by 1. - Continue (1) and (2) till complete number is covered.
Usage:
Vinculum numbers are used especially whenever we have higher digits (6, 7, 8, 9) in a number for carrying out Subtraction, Multiplication, Division, etc. Like any other number, Vinculum Number is a hypothetical number used in Vedic Mathematics to make calculations faster.
Similarly normal numbers can be converted to
Vinculum (Bar) numbers. (i.e. converting bigger digits like 6,7,8,9 to smaller
digits like 1,2,3,4.) (watch below Video “Vedic Mathematics -2
(Vinculum 2of3)”).
Similar process can be followed from R –> L
- Convert normal digit in Bar digit (By taking its 10’s complement)
- BUT Increment the previous digit by 1.
Let’s take few examples in which we don’t want given number to have any digit greater than 5.
(Convert Greater Digits like 6, 7, 8, 9 to smaller Vinculum Digits like 1, 2, 3, 4 using Vinculum process.)
Normal number –> Vinculum (Bar) number (Check the video)
R –> L
- Find 1st digit > 5, take its 10’s complement with a Bar over it.
- (a).If next digit is >= 5, take its 9’s
complement with a Bar over it & continue this till a digit < 5 is
obtained.
(b). Increment <5 digit by 1. - Continue (1) and (2) till complete number is covered.
Subtraction using Vedic Mathematics:
Most of us have difficulty and uncertainty whenever we need to do subtraction, especially when it comes to subtraction of larger digits from smaller digits.
Vedic Mathematics’s Vinculum process can also be used for Subtracting 2 numbers. (How ?? watch my below Video “Vedic Mathematics -1 (Vinculum 1of3)”).
Process (R –> L):
- If bigger digit is to be subtracted from smaller digit, then just perform Bigger digit – Smaller digit and write bar over it.
- If smaller digit is to be subtracted from bigger digit then carry usual method.
- In final answer (Vinculum number) convert to normal number using Vinculum process.
Multiplication and Division examples in Vedic Maths which uses Vinculum Process will be seen in respective topics.
3. Quotient and Remainders:
This is another important concept of Vedic Mathematics. We will follow the below formula and the concept that Remainder is ALWAYS < Divisor.
Dividend = Quotient x Divisor + Remainder
Examples:
33 ÷ 6 = 5/3 … where 5 -> Quotient & 3
-> Remainder
34 ÷ 6 = 5/4 … where 5 -> Quotient & 4 -> Remainder
35 ÷ 6 = 5/5 … where 5 -> Quotient & 5 -> Remainder
36 ÷ 6 = 6/0 … where 6 -> Quotient & 0 -> Remainder
So on observation we can say that the Remainder can never be >= Divisor.
But in Vedic mathematics in some examples depending on some criteria we need to play with Quotients and Remainders i.e. use Remainder >= Divisor for carrying out the process. But the in final answer Remainder can never be >= Divisor.
Thus for carrying out the process above examples (from bottom to top) can also be written as: ( ( watch my below Video “Vedic Mathematics -4 (Playing with Quotients and Remainders)”).)
36 ÷ 6 = 5/6 OR 4/12 OR 3/18 and so on.
35 ÷ 6 = 4/11 OR 3/17 and so on.
And vice versa for calculating final answer(Remainder can never be >= Divisor).
If we obtained Remainder(R) which is >=
Divisor(D), we divide R by D and corresponding obtained quotient is added with
obtained Q and new remainder becomes our R.
Lets say we obtained Q=4 & R=12, and D used was 11. As R > D, Apply
above rule.
So Actuals are Q= 5 and R=1. Q=4 & R=18, and D used was 6. As R > D,
Apply above rule. So Actuals are Q= 7 and R=0.aring is caring!
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